Remark 2.4 (Classic trisections) 
If all of the compression bodies \(C_{(i)}^j\)  have empty negative boundary, each \(Y^\pm _{C_{(i)}^0,C_{(i)}^1;g}\)  is a handlebody and it follows that \(X\)  is closed. Here, the definition above agrees with the original definition of a trisection when \(X\)  is closed [1]}.
If for all \(i\)  we have that \(C_{(i)}^0\)  and \(C_{(i)}^1\)  are compression bodies determined by the same loops with connected negative boundary or \(Y^\pm _{C_{(i)}^0,C_{(i)}^1;g}\)  are determined by the curves in Figure 5 of [2]}, then this decomposition is the same as a trisection of a 4-manifold with boundary in [1]}, [4]}, [2]}.
Such trisections induce an open book decomposition on \(\partial X\)  with binding a \(b\) -component link.
We will refer to all the above as classical trisections, among them we refer to the ones with \(\partial X\ne \emptyset \)  as relative trisections. We will sometimes wish to distinguish whether or not the binding of the open book \(\partial X\)  is empty; in these cases we will simply write \(b=0\)  or \(b>0\) , referring to the number of boundary components on the relative trisection diagram.
The most general trisection, or \(\star \) -trisection, is when \(C_{(i)}^0\)  is not the same as \(C_{(i)}^1\)  for some \(i\) . The following section is dedicated to present different ways of thinking about the sectors of \(\star \) -trisections.

